The main results of their work are regularity and counting lemmas for k-uniform hypergraphs for an arbitrary k ≥ 2.
developed a notion of hypergraph regularity, including an idea of pseudorandomness for hypergraphs, which has applications to graph theory as well as other fields such as number theory and combinatorial geometry. However, dealing with hypergraphs is difficult because of their complex nature, and an extension of graph regularity to hypergraphs has not been easily found. Although graph theory is appropriate for describing binary relations on objects, the more general structure of hypergraphs is the appropriate tool for multiple relations.
A powerful tool in graph theory is Szemerédi's regularity lemma, which roughly states that any graph splits into small graphs that are in some sense pseudorandom, which renders arbitrary graphs more manageable.
